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We prove that a word hyperbolic group whose Gromov boundary properly contains a 2-sphere cannot admit a projective Anosov representation into Sp₂₌ (C), m N. We also prove that a word hyperbolic group which admits a projective Anosov representation into Sp₂₌ (R) is virtually a free group or virtually a surface group, a result established indepedently by Dey-Greenberg-Riestenberg.
Pozzetti et al. (2023) studied this question.